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New York hero bumps establishment's elected bench-warmer (10)

SUB (hero, sandwich in US) replacing IN (elected) in INSTITUTE, (establishment)
In AU, a bench-warmer can be a backbencher, or an ineffective politician.

Change regular payments to a bird picked up - in this? (10)

SUBS TIT UTE

UTE = Antipodean utility vehicle or pick-up truck

Seti and Tut were agitated on the carriage back; one would overthrow the other (10)

Fashionable tutu is best for reserve (10)

Replace confused tuba players with native America. (10)

Tuba players are TUBISTS

TUBISTS* UTE

https://en.wikipedia.org/wiki/Ute_people

Eunuch has no right using ties to Brutus) 10)

Eunuchs replaced soldiers guarding the harem, I wonder why

Eunuch needs a tweak, it doesn't work

Eunuch using ties to Brutus, or not (10)

@25 should, of course, read native American.

Another version

Eunuch using ties to Brutus stripped of gold (10)